Minor-Closed Graph Classes with Bounded Layered

Vida Dujmovi´c ‡ † Gwena¨elJoret § Pat Morin ∗ David R. Wood ¶

19th October 2018; revised 4th June 2020

Abstract

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.

1 Introduction

Pathwidth and are graph parameters that respectively measure how similar a given graph is to a path or a . These parameters are of fundamental importance in structural , especially in Roberston and Seymour’s graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23].

Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition.

‡School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada ([email protected]). Research supported by NSERC and the Ontario Ministry of Research and Innovation. §D´epartement d’Informatique, Universit´eLibre de Bruxelles, Brussels, Belgium ([email protected]). Supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. †Department of Computer Science, University of California, Irvine, California, USA ([email protected]). Supported in part by NSF grants CCF-1618301 and CCF-1616248. ∗School of Computer Science, Carleton University, Ottawa, Canada ([email protected]). Research supported by NSERC. ¶School of Mathematics, Monash University, Melbourne, Australia ([email protected]). Research supported by the Australian Research Council.

1 The purpose of this paper is to characterise the minor-closed graph classes with bounded layered pathwidth.

1.1 Definitions

Before continuing, we define the above notions. A tree decomposition of a graph G is a

collection (Bx ⊆ V (G): x ∈ V (T )) of subsets of V (G) (called bags) indexed by the nodes of a tree T , such that:

(i) for every edge uv of G, some bag Bx contains both u and v, and

(ii) for every vertex v of G, the {x ∈ V (T ): v ∈ Bx} induces a non-empty connected subtree of T .

The width of a tree decomposition is the size of the largest bag minus 1. The treewidth of a graph G, denoted by tw(G), is the minimum width of a tree decomposition of G.

A path decomposition is a tree decomposition in which the underlying tree is a path. We

denote a path decomposition by the corresponding sequence of bags (B1,...,Bn). The pathwidth of G, denoted by pw(G), is the minimum width of a path decomposition of G.

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. A class of graphs G is minor-closed if for every G ∈ G, every minor of G is in G.

A layering of a graph G is a partition (V0,V1,...,Vt) of V (G) such that for every edge vw ∈ E(G), if v ∈ Vi and w ∈ Vj then |i − j| 6 1. Each set Vi is called a layer. For example, for a vertex r of a connected graph G, if Vi is the set of vertices at distance i from r, then (V0,V1,... ) is a layering of G, called the bfs layering of G starting from r. Dujmovi´cet al. [19] introduced the following definition. The layered width of a tree

decomposition (Bx : x ∈ V (T )) of a graph G is the minimum integer ` such that, for some layering (V0,V1,...,Vt) of G, each bag Bx contains at most ` vertices in each layer Vi. The layered treewidth of a graph G, denoted by ltw(G), is the minimum layered width of a tree decomposition of G. Bannister et al. [1] defined the layered pathwidth of a graph G, denoted by lpw(G), to be the minimum layered width of a path decomposition of G.

For a graph parameter f (such as treewidth, pathwidth, layered treewidth or layered pathwidth), a graph class G has bounded f is there exists a constant c such that f(G) 6 c for every graph G ∈ G.

1.2 Examples and Applications

Several interesting graph classes have bounded layered treewidth (despite having unbounded treewidth). For example, Dujmovi´cet al. [19] proved that every has layered treewidth at most 3, and more generally that every graph with Euler g has layered treewidth at most 2g + 3. Note that layered treewidth and layered pathwidth are not minor-closed parameters (unlike treewidth and pathwidth). In fact, several graph classes

2 that contain arbitrarily large minors have bounded layered treewidth or bounded layered pathwidth. For example, Dujmovi´c,Eppstein, and Wood [15] proved that every graph that can be drawn on a surface of Euler genus g with at most k crossings per edge has layered treewidth at most 2(2g + 3)(k + 1). Even with g = 0 and k = 1, this family includes graphs with arbitrarily large clique minors. Map graphs have similar behaviour [15].

Bannister et al. [1] identified the following natural graph classes1 that have bounded layered pathwidth (despite having unbounded pathwidth): every squaregraph has layered pathwidth 1; every bipartite has layered pathwidth 1; every outerplanar graph has layered pathwidth at most 2; every has layered pathwidth at most 2; and every unit disc graph with clique number k has layered pathwidth at most 4k.

Part of the motivation for studying graphs with bounded layered treewidth or pathwidth is that such graphs have several desirable properties. For example, Norin proved that every √ n-vertex graph with layered treewidth k has treewidth less than 2 kn (see [19]). This leads to a very simple proof of the Lipton-Tarjan separator theorem. A standard trick leads to an √ upper bound of 11 kn on the pathwidth (see [15]).

Another application is to stack layouts (or book embeddings), queue layouts and track layouts. Dujmovi´cet al. [19] proved that every n-vertex graph with layered treewidth k has track- and queue-number O(k log n). This leads to the best known bounds on the track- and queue-number of several natural graph classes2. For graphs with bounded layered pathwidth, the dependence on n can be eliminated: Bannister et al. [1] proved that every graph with layered pathwidth k has track- and queue-number at most 3k. Similarly, Dujmovi´c,Morin, and Yelle [20] proved that every graph with layered pathwidth k has stack-number at most 4k.

Graph colouring is another application area for layered treewidth. Esperet and Joret [22] proved that every graph with maximum ∆ and Euler genus g is (improperly) 3- colourable with bounded clustering, which means that each monochromatic component has size bounded by some function of ∆ and g. This resolved an old open problem even in the planar case (g = 0). The clustering function proved by Esperet and Joret [22] is roughly O(∆32∆ 2g ). While Esperet and Joret [22] made no effort to reduce this function, their method will not lead to a sub-exponential clustering bound. On the other hand, Liu and Wood [30] proved that every graph with layered treewidth k and maximum degree ∆ is 3-colourable with clustering O(k19∆37), which was recently improved to O(k3∆2) by Dujmovi´c,Esperet, Morin, Walczak, and Wood [16]. In particular, every graph with Euler genus g and maximum degree ∆ is 3-colorable with clustering O(g3∆2). This greatly improves upon the clustering bound of Esperet and Joret [22]. Moreover, the proofs in [16, 30] are relatively simple, avoiding many technicalities that arise when dealing with graph embeddings. These results highlight the utility of layered treewidth as a general tool.

1A squaregraph is a graph that has an embedding in the plane in which each bounded face is a 4- and each vertex either belongs to the unbounded face or has degree at least 4. A Halin graph is a planar graph obtained from a tree T with at least four vertices and with no vertices of degree 2 by adding a cycle through the leaves of T in the clockwise order determined by a plane embedding of T . For a set P of points in the plane, the unit disc graph G of P has vertex set P , where vw ∈ E(G) if and only if dist(v, w) 6 1. 2Subsequent to the submission of this paper, improved bounds on the queue-number have been obtained [17]. Still it is open whether graphs of bounded layered treewidth have bounded queue-number.

3 1.3 Characterisations

We now turn to the question of characterising those minor-closed classes that have bounded treewidth. The key example is the n×n grid graph, which has treewidth n. Indeed, Robertson and Seymour [34] proved that every graph with sufficiently large treewidth contains the n × n grid as a minor. The next theorem follows since every planar graph is a minor of some grid graph. Several subsequent works have improved the bounds [6–8, 14, 29, 36]. Theorem 1 (Robertson and Seymour [34]). A minor-closed class has bounded treewidth if and only if some planar graph is not in the class.

An analogous result for pathwidth holds, where the complete is the key example

(the analogue of grid graphs for treewidth). Let Th be the complete binary tree of height h (the rooted tree in which each non-leaf vertex has exactly two children, and the distance from h the root to each leaf vertex equals h). It is well known and easily proved that pw(Th) = d 2 e, and every forest is a minor of some complete binary tree. Robertson and Seymour [33] proved the following characterisation. Theorem 2 (Robertson and Seymour [33]). A minor-closed class has bounded pathwidth if and only if some forest is not in the class.

Note that Bienstock, Robertson, Seymour, and Thomas [2] proved the following quantitatively stronger result: for every forest T with |V (T )| > 2 every graph containing no T minor has pathwidth at most |V (T )| − 2.

Now consider layered analogues of Theorems1 and2. A graph G is apex if G − v is planar for some vertex v. Define the n × n to be the obtained from the n × n grid by adding one dominant vertex v. (Here a vertex is dominant if it is adjacent to every other vertex in the graph.) The n × n pyramid has treewidth n + 1 and layered treewidth n+2 at least 3 , since every layering uses at most three layers. Pyramids are ‘universal’ apex graphs, in the sense that every apex graph is a minor of some pyramid graph (since every planar graph is a minor of some grid graph). Dujmovi´cet al. [19] proved the following characterisation. Theorem 3 (Dujmovi´cet al. [19]). A minor-closed class has bounded layered treewidth if and only if some apex graph is not in the class.

Theorem3 generalises the above-mentioned result that graphs of bounded Euler genus have bounded layered treewidth.

A graph G is an apex-forest if G − v is a forest for some vertex v. The following analogue of Theorem3 is the main result of this paper. Theorem 4. A minor-closed class has bounded layered pathwidth if and only if some apex-forest is not in the class.

Dujmovi´cet al. [19] noted that Theorem3 implies Theorem1. Similarly, we now show 3

3While Theorem4 implies Theorem2, we emphasise that our proof of Theorem4 relies on the above- mentioned quantitative version of Theorem2 by Bienstock et al. [2]. Similarly, the proof of Theorem3 uses the structure theorem of Robertson and Seymour [35] and thus relies on Theorem1.

4 that Theorem4 implies Theorem2. Let T be a forest, and let G be a graph with no T minor. Let T + be the apex-forest obtained from T by adding a dominant vertex v. Let G+ be the graph obtained from G by adding a dominant vertex x. Suppose for the sake of contradiction that G+ contains a T +-minor. A T +-minor in G+ can be described by a mapping from the vertices of T + to vertex-disjoint trees in G+ such that whenever two vertices in T + are adjacent, the corresponding two trees induce a connected subgraph of G. From this mapping, remove two (not necessarily distinct) trees, the image of v and the tree (if it exists) that contains x. If the tree that contains x was the image of a vertex w in T , then instead map w to the tree that was the image of v. The resulting mapping describes a T -minor in G, as claimed. This contradiction shows that G+ is T +-minor-free. By Theorem4, G+ has layered pathwidth at most c = c(T +). Since G+ has radius 1, at most three layers are used. Thus G+ and G have pathwidth less than 3c.

Layered treewidth is closely related to the notion of ‘local treewidth’, which was first introduced by Eppstein [21] under the guise of the ‘treewidth-diameter’ property. A graph class G has bounded local treewidth if there is a function f such that for every graph G in G, for every vertex v of G and for every integer r > 0, the subgraph of G induced by the vertices at distance at most r from v has treewidth at most f(r). If f(r) is a linear function, then G has linear local treewidth. See [10, 11, 21, 23, 24] for results and algorithmic applications of local treewidth. Dujmovi´cet al. [19] observed that if some class G has bounded layered treewidth, then G has linear local treewidth. On the other hand, bounded layered treewidth is a stronger property than bounded or linear local treewidth.

Local pathwidth is defined similarly to local treewidth. A graph class G has bounded local pathwidth if there is a function f such that for every graph G in G, for every vertex v of G and for every integer r > 0, the subgraph of G induced by the vertices at distance at most r from v has pathwidth at most f(r). The observation of Dujmovi´cet al. [19] extends to the setting of local pathwidth; see Lemma9 below.

Theorem4 is extended to capture local pathwidth by the following theorem, which also provides a structural description in terms of a tree decomposition with certain properties that we now introduce. If T is a tree indexing a tree decomposition of a graph G, then for each vertex v of G, let T [v] denote the subtree of T induced by those nodes corresponding to bags that contain v. Thus T [v] is non-empty and connected. Say that a tree decomposition of a graph G is (w, p)-good if its width is at most w and, for every v ∈ V (G), the subtree T [v] has pathwidth at most p. We illustrate this definition with two examples. Let T be a tree, rooted at some vertex. For each node x of T , introduce a bag Bx consisting of x and its parent node (or just x if x is the root). Then (Bx : x ∈ V (T )) is a tree decomposition of T with width 1. Moreover, for each vertex v, the subtree T [v] is a star, which has pathwidth 1. Thus every tree has a (1, 1)-good tree decomposition. Now, consider an outerplanar triangulation G. Let T be the weak dual tree (ignoring the outerface). For each node x of

T , let Bx be the set of three vertices on the face corresponding to x. Then (Bx : x ∈ V (G)) is a tree decomposition of G with width 2. Moreover, for each vertex v of G, the subtree T [v] is a path, which has pathwidth 1. Thus every outerplanar graph has a (2, 1)-good tree decomposition (since every outerplanar graph is a subgraph of an outerplanar triangulation). These constructions are generalised via the following theorem, which immediately implies Theorem4.

5 Theorem 5. The following are equivalent for a minor-closed class G:

(1) some apex-forest graph is not in G,

(2) G has bounded local pathwidth,

(3) G has linear local pathwidth.

(4) G has bounded layered pathwidth,

(5) there exist integers w and p, such that every graph in G has a (w, p)-good tree decompo- sition.

Here is some intuition about property (5). Suppose that G excludes some apex-forest graph as a minor. Since every apex-forest graph is planar, by Theorem1, the graphs in G have bounded treewidth. Thus we should expect that the tree decompositions in (5) have bounded width. Moreover, if G has bounded layered pathwidth, then G[N(v)] has bounded pathwidth for each vertex v in each graph G ∈ G. Property (5) takes this idea further, and says that each subtree T [v] has bounded pathwidth, which implies that G[N(v)] has bounded pathwidth (since the width of the tree decomposition is bounded; see Lemma6).

In Section2 we prove (5) = ⇒ (4) =⇒ (3) =⇒ (2) =⇒ (1). In Section3 we close the loop by proving (1) =⇒ (5). This proof uses a recent characterisation by Dang [9] of the unavoidable minors in 3-connected graphs of large pathwidth.

Throughout the proof we use the following ‘universal’ apex-forest graph. Let Qk be the graph obtained from the complete binary tree Tk by adding one dominant vertex. Note that k k+4 pw(Qk) = d 2 e + 1 and the layered pathwidth of Qk is at least 6 , since every layering of Qk uses at most three layers. Since every forest is a minor of some complete binary tree, every apex-forest graph is a minor of some Qk.

2 Downward Implications

We start with a few simple but useful lemmas.

Lemma 6. If a graph G has a tree decomposition of width k indexed by a tree of pathwidth p, then G has pathwidth at most (p + 1)(k + 1) − 1.

Proof. Let (Bx : x ∈ V (T )) be a tree decomposition of G of width k. Let (C1,...,Cn) be a path decomposition of T of width p. For i ∈ {1, . . . , n}, let D := S B . Then i x∈Ci x (D1,...,Dn) is a path decomposition of G of width (p + 1)(k + 1) − 1 (since |Ci| 6 p + 1 and |Bx| 6 k + 1).

Lemma 7. Let T1 and T2 be subtrees of a tree T , such that T = T1 ∪ T2. Then

pw(T ) + 1 6 (pw(T1) + 1) + (pw(T2) + 1).

6 Proof. Let (B1,...,Bs) be a path decomposition of T1 with bag size at most pw(T1) + 1. Each component of T − V (T1) is contained in T2 and therefore has a path decomposition with bag size at most pw(T2) + 1. For each such component J of T − V (T1), there is exactly one vertex v in T1 adjacent to some vertex in J (otherwise T would contain a cycle consisting of two edges between a path in T1 and a path in J). Say v is in bag Bi. We say J attaches at v and at Bi. By doubling bags in the path decomposition of T1, we may assume that distinct components of T − V (T1) attach at distinct Bi. For each component J of T − V (T1), if (D1,...,Dt) is a path decomposition of J with bag size at most pw(T2) + 1, then replace Bi by (Bi ∪ D1,...,Bi ∪ Dt). We obtain a path decomposition of T with bag size at most (pw(T1) + 1) + (pw(T2) + 1). The result follows.

Corollary 8. Let T1,...,Tk be subtrees of a tree T , such that T = T1 ∪ · · · ∪ Tk. Then

k X pw(T ) + 1 6 (pw(Ti) + 1). i=1

We now prove the downward implications in Theorem5. First note that (2) implies (1),

since if every graph in G has local pathwidth at most k, then the apex-forest graph Q6k is not in G. It is immediate that (3) implies (2). That (4) implies (3) is the above-mentioned observation of Dujmovi´cet al. [19] specialised for pathwidth. We include the proof for completeness.

Lemma 9. Let G be a class of graphs such that every graph in G has layered pathwidth at most k. Then G has linear local pathwidth with binding function f(r) = (2r + 1)k − 1.

Proof. For a graph G ∈ G, let (B1,...,Bs) be a path decomposition of G with layered width k, with respect to some layering (V0,V1,...,Vt). Let v be a vertex in Vi. Let r be a positive integer. Let H be the subgraph of G induced by the vertices at distance at most r from

v. Thus V (H) ⊆ Vi−r ∪ Vi−r+1 ∪ · · · ∪ Vi+r. Each bag Bj contains at most k vertices in each layer. Hence (B1 ∩ V (H),...,Bs ∩ V (H)) is a path decomposition of H with at most (2r + 1)k vertices in each bag. Therefore G has linear local pathwidth with binding function f(r) = (2r + 1)k − 1.

The next lemma shows that (5) implies (4).

Lemma 10. If a graph G has a (w, p)-good tree decomposition, then

lpw(G) 6 w(p + 1)(w + 1).

Proof. Let T = (Bx : x ∈ V (T )) be a tree decomposition of G with width w, such that pw(T [v]) 6 p for each vertex v of G. Since adding edges does not decrease the layered pathwidth, we may add edges to G between two non-adjacent vertices in the same bag of T . Now each bag is a clique, and G is chordal with maximum clique size w + 1. Let

(V0,V1,...,Vt) be a bfs layering in G. That is, Vi is the set of vertices in G at distance i from some fixed vertex r of G. In particular, V0 = {r}.

Consider a component H of G[Vi] for some i > 1. Let CH be the set of vertices in Vi−1 adjacent to at least one vertex in H. Since G is chordal, CH is a clique of size at most w

7 (see [18, 28]), called the parent clique of H. Define T := S T [u]. Since C is a clique, H u∈CH H which is contained in a single bag of T , there is a node x of T such that x ∈ T [u] for each

u ∈ CH . Thus TH is a (connected) subtree of T . Moreover, TH is the union of at most w subtrees, each with pathwidth at most p. Thus pw(TH ) + 1 6 w(p + 1) by Corollary8. Let Hˆ := G[V (H) ∪ CH ].

We now prove that TH := (Bx ∩ V (Hˆ ): x ∈ V (TH )) is a tree decomposition of Hˆ . We first prove condition (ii). For a vertex v of CH , the set of bags of TH that contain v is precisely those indexed by nodes in T [v], which is non-empty and connected, by assumption. Now, consider a vertex v in H. Let w be the neighbour of v on a shortest vr-path in G. Thus w

is in CH . Since vw is an edge, v and w appear in a common bag of T , which corresponds to a node in TH (since that bag contains w). Hence TH [v] is non-empty. We now prove that TH [v] is connected. Let B1 and B2 be distinct bags of TH containing v. Let P be the B1B2-path in T . Since T [v] is connected, v is in the bag associated with each node in P . To conclude that TH [v] is connected, it remains to prove that P ⊆ TH . By construction, some vertex w1 is in B1 ∩ CH and some vertex w2 is in B2 ∩ CH . Since w1 and w2 are adjacent, the bag associated with each node in P contains w1 or w2. Hence P ⊆ TH and TH [v] is connected. This proves condition (ii). Now we prove condition (i). Since CH is contained in some bag of TH , condition (i) holds for each edge with endpoints in CH . For each edge vw with v ∈ V (H) and w ∈ CH , v and w are in a common bag Bx of T , implying x is in TH (since Bx contains w), as desired. Finally, consider an edge uv with u, v ∈ V (H). Suppose on the contrary that u and v have no common neighbour in CH . By construction, u has a neighbour w1 in CH , and v has a neighbour w2 in CH . Thus w1 6= w2. Since CH is a clique, w1 and w2 are adjacent. Since uw2 6∈ E(G) and vw1 6∈ E(G), the 4-cycle (u, w1, w2, v) is chordless, and G is not chordal, which is a contradiction. Hence u and v have a common

neighbour w in CH . Thus {u, v, w} is a triangle in G, which is in a common bag of T , and therefore in a common bag of TH , implying that u and v are in a common bag of TH . This proves condition (i) in the definition of tree decomposition. Therefore TH is a tree decomposition of Hˆ . By construction, it has width at most w. ˆ Since pw(TH ) + 1 6 w(p + 1) and TH indexes a tree decomposition of H with width at most ˆ w, by Lemma6, pw( H) 6 (w(p + 1) + 1)(w + 1) − 1. We now construct a path decomposition of G with layered width at most w(p + 1)(w + 1)

with respect to layering (V0,V1,...,Vt). Let Gi := G[V0 ∪ V1 ∪ · · · ∪ Vi]. We now prove, by induction on i, that Gi has a path decomposition with layered width at most w(p + 1)(w + 1) with respect to layering (V0,V1,...,Vi). This claim is trivial for i = 0. Now assume that (B1,...,Bq) is a path decomposition of Gi−1 with layered width at most w(p + 1)(w + 1) with respect to layering (V0,V1,...,Vi−1). For each component H of G[Vi], there is a bag Bj that contains CH ; pick one such bag and call it the parent bag of H. By doubling the bags, we may assume that distinct components of G[Vi] have distinct parent bags. Now, for each component H of G[Vi] with parent bag Bj, if (D1,...,Ds) is a path decomposition of Hˆ with width w(p + 1)(w + 1) − 1, then replace Bj by (Bj ∪ D1,...,Bj ∪ Ds). Doing this for each component of G[Vi] produces a path decomposition of Gi with layered width at most w(p + 1)(w + 1) with respect to layering (V0,V1,...,Vi). In particular, we obtain a path decomposition of G with layered width at most w(p + 1)(w + 1) with respect to

layering (V0,V1,...,Vt).

8 3 Proof that (1) implies (5)

The goal of this section is to show that if a graph G excludes some apex-forest graph H as a minor, then G has a (w, p)-good tree decomposition for some w = w(H) and p = p(H).

Since every apex-forest graph is a minor of some Qk, it suffices to prove this result for H = Qk, in which case we denote w = w(k) and p = p(k). We will be working with two related trees S and T and one graph G. To help the reader keep track of things we use variables a, b, and c as names for nodes of S and T and variables v, x, y, and z to refer to vertices of G.

We now give an outline of the proof. First, we show that a recent result by Dang [9] implies

that every 3-connected graph G with no Qk minor has pathwidth at most w = w(k). Thus, in this case, G has a (w, 1)-good tree decomposition. Next we deal with cut vertices by showing that if each block of a graph G has a (w, p)-good tree decomposition, then G has a (w, p + 1)-good tree decomposition.

Therefore, the main difficulty is to show that every 2-connected graph G with no Qk minor has a (w, p)-good tree decomposition (Ba : a ∈ V (T )). By the result of Bienstock et al. [2] described in the introduction, if pw(T [v]) > 2h+1 − 3 for some v ∈ V (G) then T [v] contains

a Th minor. For sufficiently large h, we then construct a Qk minor (from the Th minor in T [v]) in which v plays the role of the apex vertex.

To construct the tree decomposition (Ba : a ∈ V (T )) we use two tools: An SPQR-tree, S, represents a graph G as a collection of subgraphs (S- and R-nodes) that are joined at 2-vertex cutsets (P-nodes). These subgraphs consist of cycles (S-nodes) and 3-connected graphs (R-nodes). Cycles have pathwidth 2 and, by the result of Dang discussed above, the 3-connected graphs have pathwidth at most w = w(k). Replacing the S- and R-nodes of the SPQR-tree with these path decompositions produces the tree T in our tree decomposition.

To show that this tree decomposition is (w, p)-good, we first show that if T [v] contains a subdivision of a sufficiently large complete binary tree, then the SPQR-tree S also contains a subdivision of a large complete binary tree all of whose nodes have subgraphs that contain

v. Using this large binary tree in S we then piece together a subgraph of G that has a Qk minor in which v is the apex vertex.

3.1 Dang’s Result

First we show how the following result of Dang [9] implies that every 3-connected graph

with no Qk minor has pathwidth at most w = w(k).

Theorem 11 (Dang [9, Theorem 1.1.5]). Let P be a graph with two distinct vertices u1 and u2 such that P − {u1, u2} is a forest, Q be a graph with a vertex v such that Q − v is outerplanar, and R be a tree with a cycle going through its leaves in order from the leftmost leaf to the rightmost leaf so that R is planar. Then there exists a number w = w(P, Q, R) such that every 3-connected graph of pathwidth at least w has a P , Q, or R minor.

Note that R is a Halin graph, except that degree-2 vertices are allowed in the tree.

9 + To use Theorem 11 we need a small helper lemma. For every k > 0, let Tk be the graph obtained from the complete binary tree Tk of height k by adding a new vertex adjacent to the leaves. The next lemma is well known.

+ Lemma 12. For every integer k > 0, T2k contains Qk as a minor.

+ Proof. The statement is immediate for k = 0. For k > 1, partition the edges of T2k into the tree T2k and the remaining edges, which form a star centered at some vertex v. Let a1, a2, a3, a4 be the grandchildren of the root of T2k ordered from left to right. Contract the entire subtree comprised of the subtree rooted at a2, the subtree rooted at a3, and the path + from a2 to a3. Applying the same procedure recursively on the copy of T2(k−1) rooted at a1 + and the copy of T2(k−1) rooted at a4 produces Qk, as can be easily verified by induction. Corollary 13. There exists a number w = w(k) such that every 3-connected graph of

pathwidth at least w has a Qk minor.

Proof. Let P be obtained from the complete binary tree Tk by adding two dominant vertices u1 and u2. Let Q be the graph obtained from the outerplanar graph ∇k, whose weak dual is a complete binary tree of height k, by adding a dominant vertex v. Let R be the graph

obtained from T2k+1 by adding a cycle on its leaves, so that R is planar.

Then P contains Qk as a minor since P − {u2} is isomorphic to Qk. Q also contains Qk as a minor because ∇k contains a complete binary tree of height k as a subgraph. Finally, R also contains a Qk minor: Contract the cycle, then we have a complete binary tree of height 2k plus an apex vertex linked to its leaves, which contains Qk as a minor by Lemma 12. Theorem 11 implies that there exists w = w(k) such that every 3-connected graph with pathwidth at least w contains at least one of P , Q, or R as a minor and therefore contains

a Qk minor.

3.2 Dealing with Cut Vertices

A block in a graph is either a maximal 2-connected subgraph, the subgraph induced by the endpoints of a edge, or the subgraph induced by an isolated vertex.

Lemma 14. Let G be a graph, such that each block of G has a (w, p)-good tree decomposition. Then G has a (w, p + 1)-good tree decomposition.

Proof. Let C1,...,Cr be the blocks of G. For each i ∈ {1, . . . , r}, let Ti be the underlying tree in a (w, p)-good tree decompositions of Ci. We create a tree decomposition of G as follows: For each cut vertex or isolated vertex v in

G, introduce a new tree node av with Bav = {v}. In each block Ci that contains v, the tree decomposition (Ba : a ∈ V (Ti)) of Ci has at least one node a such that v ∈ Ba; make av adjacent to exactly one such node for each Ci. It is straightforward to verify that this defines a tree decomposition of G and we now argue this decomposition is (w, p + 1)-good. The resulting tree decomposition of G has width

10 Figure 1: A graph and its SPQR-tree. at most w. For each isolated vertex v ∈ V (G), the subtree T [v] consists of one node. For each cut vertex v ∈ V (G), the subtree T [v] is composed of some number of subtrees, each adjacent to av and each having a path decomposition of width at most p. We obtain a path decomposition of T [v] by concatenating the path decompositions of each subtree and adding v to every bag of the resulting path decomposition. The resulting path decomposition of T [v] has width at most p + 1.

3.3 SPQR-Trees

In this section, we quickly review SPQR-trees, a structural decomposition of 2-connected graphs used previously to characterize planar embeddings [31], to design efficient algorithms for triconnected components [26], and in efficient data structures for incremental [12, 13].

Let G be a 2-connected graph. An SPQR-tree S of G is a tree in which each node a ∈ V (S) is associated with a minor Ha of G. For any S- or R-node a of S, Ha is a simple graph. If a is a P-node, on the other hand, then Ha is a dipole graph having two vertices and at least two parallel edges. In all cases, Ha is a minor of G. For a P-node a in which Ha contains vertices x and y and t parallel edges, this means that G contains t internally disjoint paths from x to y. For each node a of S each edge xy ∈ E(Ha) is classified either as a virtual edge or a real edge. An SPQR-tree S is defined recursively as follows (see Figure1): 4

1. If G is a cycle, then S consists of a single node a (an S-node) in which Ha = G and all edges of Ha are real.

2. If G is 3-connected, then S consists of a single node a (an R-node) in which Ha = G and all edges of Ha are real.

4This definition includes P-nodes consisting of only two virtual edges, which some works exclude because they are unnecessary. However, their inclusion simplifies some of our analysis.

11 3. Otherwise G has a cutset {x, y} such that x and y each have degree at least 3. Then let C1,...,Cr, r > 2, be the connected components of G − {x, y}. For each i ∈ {1, . . . , r}, let G˜i be G[V (Ci) ∪ {x, y}] along with the additional edge xy, if not already present. (The edge xy is treated as a real edge so that the node ai mentioned below exists.) Because of the inclusion of xy, each G˜i is 2-connected, so each has an SPQR-tree Si. Then an SPQR-tree for G is obtained by creating a node a (a P-node) with Ha being a dipole graph with vertices x and y and having r virtual edges joining x and

y. In addition to these virtual edges, Ha contains the real edge xy if xy ∈ E(G). The construction and the fact that xy is an edge in each G˜i imply that, for each

i ∈ {1, . . . , r}, there exists exactly one node ai in Si such that xy is a real edge in Hai . To complete S, make a adjacent to each of a1, . . . , ar, and make xy a virtual edge in

each of Ha1 ,...,Har .

Let S be an SPQR-tree of a 2-connected graph G. For each node a of S, let Er(Ha) denote the set of real edges in Ha and let Ev(Ha) denote the multiset of virtual edges in Ha. For a connected subtree S0 of S, define G[S0] to be the subgraph of G whose vertex set 0 S 0 S is V (G[S ]) = a∈V (S0) V (Ha) and whose edge set is E(G[S ]) = a∈V (S0) Er(Ha). For a vertex v ∈ V (G), let S[v] := S[{a ∈ V (S): v ∈ V (Ha)}], which is called the subtree of S induced by v. We make use of the following properties of S:

1. Every R-node and S-node is adjacent only to P-nodes and no two P-nodes are adjacent.

2. The degree of every node a is equal to the number of virtual edges in Ha.

3. For every vertex v ∈ V (G), S[v] is connected.

4. If a is an R-node or S-node, then Ha is a simple graph; that is, Ha contains no parallel edges.

5. If a P-node a has degree 2 and both its neighbors are S-nodes then Ha has a real edge.

6. For each node a of S and component S0 of S − {a}, G[S0] is connected.

7. For each xy ∈ E(G) there is exactly one node a of S for which xy is a real edge in Ha.

3.4 The Good Tree Decomposition

To obtain our good tree decomposition (Ba : a ∈ V (T )) of a 2-connected graph G we start with an SPQR-tree S for G. For each R-node or S-node a of S, let (Bc : c ∈ V (Pa)) be a minimum-width path decomposition of Ha. The tree T includes the path Pa for each R-node and S-node a of S. We say that the node a of S generates the nodes in the path Pa and that each node in Pa is generated by a. Each S- or R-node a is adjacent to some set of P-nodes in S. For each such P-node b whose dipole graph Hb has vertices x and y, the edge xy is a (virtual) edge in Ha and therefore x and y appear in some common bag Bc with c ∈ V (Pa). Add c and b as vertices in T and add the edge bc to T . Add b as a node of T and add the edge bc to T . This defines the tree T in the tree decomposition.

12 We now describe the contents of T ’s bags. Each P-node a of S becomes a node in T whose bag contains only the two vertices of Ha. Every node a in T that is generated by an S- or 0 R-node a of S is a node in some path decomposition of Ha0 and already has an associated bag Ba that it inherits from this path decomposition.

It is straightforward to verify that (Ba : a ∈ V (T )) is indeed a tree decomposition of G: For each vertex v ∈ V (G), the connectivity of the subtree T [v] follows from Property 3 of SPQR-trees and the equivalent property for the path decompositions that include v. Each edge xy of G appears as an edge in Ha for at least one node a of S and therefore x and y appear in a common bag in the path decomposition of Ha.

Each bag Ba of (Ba : a ∈ V (T )) either has size in {2, 3} (when a is generated by a P-node or an S-node) or it has size at most w(k) + 1 where w(k) is the function in Corollary 13 (when a is generated by an R-node). Thus, (Ba : a ∈ V (T )) is a tree decomposition of G whose width is upper bounded by a function of k. It remains to show that, for every v ∈ V (G), T [v] has pathwidth that is upper bounded by a function of k.

In the remainder of this section, we fix G to be a 2-connected graph, S to be an SPQR-tree of G, and (Ba : a ∈ V (T )) to be a tree decomposition of G obtained using the procedure described above. 2h+1 Lemma 15. For every integer h > 1, if T [v] has pathwidth greater than 2 − 3, then S[v] contains a subdivision of Th.

Proof. In the following, a binary tree is a tree rooted at a degree-2 node, such that every other node has degree in {1, 2, 3}. In a binary tree, the root and every degree 3 node is called a branching node. Every branching node and every leaf is a distinctive node. We use the convention that all binary trees are ordered, possibly arbitrarily, so that we can distinguish between the left and right child of a branching node. For a node a in a binary tree T , we denote by aˆ the subtree rooted at a; that is, the subtree of T induced by the set of nodes that have a as an ancestor, including a itself.

Recall, from the result of Bienstock et al. [2] discussed in the introduction, that if T [v] has 2h+1 0 0 pathwidth greater than 2 − 3 then T contains a subdivision T of T2h. Note that T does not immediately imply the existence of Th in S[v] since two or more distinctive nodes of T 0 may have been generated by the same node of S. Label each node of T 0 with the node of S that generated it. Recall that each node a in S generates a path in T . So a maximal subset of nodes of T 0 with a common label induces a path in T 0.

0 00 00 We claim that T contains a subdivision T of Th such that no two distinctive nodes of T have the same label. We establish this claim by induction on h: If h = 0 then the claim is trivial. Otherwise, let a be the root of T 0 and let a0 and a00 be the highest branching nodes in the left and right subtrees of aˆ, respectively. Let a1 and a2 be the highest distinctive 0 nodes in the left and right subtrees of aˆ , respectively, and let a3 and a4 be the highest distinctive nodes in the left and right subtrees of aˆ00, respectively. Since each label induces 0 a path in T , at least one of {a1, a2}, say a1, and at least one of {a3, a4}, say a4, does not have the same label as a. Furthermore, since a1 and a4 are separated by a, the set of labels of nodes in aˆ1 is disjoint from the set of labels of nodes in aˆ4. Applying induction on aˆ1 and aˆ4 yields two subdivisions of Th−1 in which no two distinctive nodes have the same label.

13 Connecting these two subdivisions with the unique path from a1 to a4 yields the desired subdivision of Th in which no two distinctive nodes have the same label. Since no two distinctive nodes in T 00 have the same label, each distinctive node corresponds to a unique node of S. Thus, contracting all nodes of T 00 that share a common label yields 000 a subtree T of S[v] that is a subdivision of Th.

Thus far we have established that if T [v] has sufficiently high pathwidth, then S[v] contains a subdivision of a large complete binary tree.

Lemma 16. If S[v] contains a subdivision of T7(k+1) then G contains a Qk minor.

Proof. First we note that if S[v] contains a subdivision of T7(k+1) then S[v] contains a 0 0 subdivision T of Tk+1 such that the path between each pair of distinctive nodes in T has length at least 7.

It is convenient to work with a simplified SPQR-tree S0 and graph G0 obtained by repeating the following operation exhaustively: Consider some edge ab of S with a ∈ V (T 0) and 0 0 b 6∈ V (T ). The edge ab is associated with some virtual edge xy in Ha. In S , replace the virtual edge xy in Ha with a real edge. At the same time, remove the maximal subtree ˆb of S that contains b and not a. By Property 6 of SPQR trees, in G0 this operation is S equivalent to contracting all the real edges in c∈V (ˆb) Er(Hc) and removing any resulting parallel edges. Since the resulting graph G0 is a minor of G, this operation is safe in the 0 sense that the existence of a Qk minor in G implies the existence of a Qk minor in G. With this simplification, the tree T 0 is an SPQR-tree for the graph G0 and every virtual edge 0 is incident to v. We now turn our efforts to finding the Qk minor in G . Recall that Qk is obtained from a complete binary tree Tk by adding a dominant vertex. We begin by finding 00 0 a subdivision T of Tk+1 in G . In this subdivision, each edge of Tk+1 that joins a node to its left child is represented by a path Pµν joining a branching node µ to a distinctive node ν. 0 We show that G contains a path from v to some anchor node η of Pµν with η 6= ν, which is vertex disjoint from T 00 except for η. Furthermore, except for their common endpoint v, 00 all of these paths are disjoint. The union of T and these paths contains a Qk minor since contracting the path from each anchor node to its closest ancestor branching node produces

Qk. See Figure2. Let a be a branching node of T 0 and let b be the nearest distinctive node in one of a’s two 0 subtrees. Consider the path a = c1, c2, . . . , cr = b in T . For each i ∈ {1, . . . , r − 1}, the edge 0 cici+1 is associated with a cutset {v, xi} in G and vxi is a virtual edge in Hci and Hci+1 .

Note that this implies that, for each i ∈ {2, . . . , r}, Hci contains both vertices xi and xi−1.

We claim that, for each i ∈ {2, . . . , r − 1}, Hci contains a path Pi from xi−1 to xi that does not contain v; refer to Figure3. When ci is a P-node, this claim is trivial since, in this case, xi−1 = xi. The case in which ci is an S-node or R-node is also easy: In these cases Hci is 2-connected, therefore there is a path from xi−1 to xi that avoids v. Now note that the paths P1,...,Pr−1 are disjoint, except for each of the common endpoints xi where Pi ends and Pi+1 0 Si begins. This is because each {v, xi} is a cutset of G that separates j=1 V (Hcj ) \{v, xi} Sr from j=i+1 V (Hcj ) \{v, xi}. By concatenating P2,...,Pr−1 we obtain a path Pab from x1 ∈ V (Ha) to xr−1 ∈ V (Hb) that we call the subdivision path for nodes a and b.

14 v

Figure 2: Finding a Qk minor in the proof of Lemma 16. Distinctive nodes are indicated by black disks and anchor nodes by white circles.

a = c1 c2 c3 c4 cr−1 b = cr v v v v v v ··· x4 xr−1 xr−1 x1 x2 x2 = x3 x3 x1 xr−2

xr−1 ··· xr−2 x4

x1 x2 = x3

Figure 3: Finding a path connecting a vertex of Ha to a vertex of Hb.

Consider any branching node a of T 0. If a is a P-node then all the subdivision paths that begin or end at a vertex of Ha include the same vertex of Ha. If a is an S- or R-node, each subdivision path that begins or ends at a vertex of Ha includes a different vertex (for up to 5 3 different vertices x, y, and z). Now, since Ha is 2-connected, these three vertices are in the same component of Ha − {v}. In particular, Ha − {v} contains an edge-minimal tree that includes x, y, and z. Adding each of these trees to the union of all subdivision paths 00 produces the subdivision T of Tk+1. Next we show how to construct paths from v to anchor nodes. Let a be a branching node of 0 T , let b be the highest distinctive node in a’s left subtree and let c1 = a, . . . , cr = b be the 0 0 path in T having endpoints a and b. Thus far we have established that G [{c2, . . . , cr−1}] contains a simple path Pab from x1 to xr−1 that does not include v. Note that r is large 0 0 enough so that c5 exists. We now show that G [{c2, c3, c4, c5}] contains an apex path P 0 from v to some anchor node of Pab such that the internal vertices of P are disjoint from

5The only time a may be an S-node is when a is the root of T 0 in which case the two subdivision paths that begin at a vertex in Ha begin at the two neighbours x and y of v in the cycle Ha.

15 0 0 0 V (Pab) ∪ V (Hb). We first describe the path P in G [{c2, c3, c4, c5}] and then show that P contains no vertex of V (Hb) \{v}. There are two cases to consider:

1. ci is an R-node, for some i ∈ {2,..., 5}: Since Hci is 3-connected, there are three

paths in Hci with endpoints v and xi and no other vertices in common. Since Hci

has only two virtual edges, at least one of these paths uses only real edges in Hci . 0 This path therefore contains a subpath P joining v to some vertex of Pab (the anchor vertex) that is otherwise disjoint from Pab.

2. Otherwise, none of c2, . . . , c5 is an R-node. Property 1 of SPQR-trees implies that, for at least one i ∈ {2, 3}, ci is an S-node, ci+1 is a P-node, and ci+2 is an S-node. 0 In the simplified SPQR-tree S , c2 and c3 each have degree 2. Therefore, Property 5 0 of SPQR-trees (applied to S ) ensures that Hci+1 contains the real edge vxi+1. This 0 real edge in the version of Hci+1 that appears in S corresponds either to a real edge 0 in the version of Hci+1 that appears in S or it was introduced in S . In the former case, G contains the edge vxi+1 and we are done. In the latter case ci+1 is adjacent in S to a node d that is not in T 0 and the maximal subtree dˆ of S that contains d but 0 not ci+1 was removed from S while producing S . Recall that this is equivalent to S contracting all the real edges c∈V (dˆ) Er(Hc). Thus, the real edge in the version of 0 Hci+1 that appears in S implies the existence of a path in G from v to xi+1 whose S internal vertices appear only in c∈V (dˆ) V (Hc). The internal vertices of this path are disjoint from Pab.

0 It remains to show that P does not contain any vertices of V (Hb) \{v}. By Properties 1 and 6 of SPQR-trees, for each x ∈ V (G0) \{v}, the subtree T 0[x] of T 0 consisting only of 0 nodes a such that x ∈ V (Ha) is a star; that is, the distance between any two nodes in T [x] is at most 2. Now, since the distance between any two distinctive nodes of T 0 is at least

7, we have r > 8 and therefore Hb = Hcr has no vertex, except v, in common with any of 0 Hc2 ,...,Hc5 . Therefore, P joins v to a vertex in Pab that is not in Hb, as required. Adding the set of all apex paths to T 00 then produces a subgraph of G0 that contains a

Qk-minor.

Finally, we have all the pieces in place to complete the proof.

Proof that (1) implies (5). Let G be a graph excluding some apex-forest graph H as a minor.

As explained earlier, G contains no Qk minor for some k = k(H). We wish to show that there are w and p that depend only on k such that G has a (w, p)-good tree decomposition. By Lemma 14 we may assume that G is 2-connected.

Consider the good tree decomposition (Ba : a ∈ V (T )) of G described in Section 3.4. This decomposition has width at most w where w = w(k) is the function that appears (implicitly) 14(k+1)+1 in Theorem 11 and Corollary 13. We claim that, for each v ∈ V (G), pw(T [v]) 6 2 −3, so that this tree decomposition is (w, 214(k+1)+1 −3)-good. Otherwise, by Lemma 15, there is a vertex v ∈ V (G) such that an SPQR-tree S has a subtree S[v] that contains a subdivision of T7(k+1). Therefore, by Lemma 16, G contains a Qk minor, contradicting the supposition that G has no Qk minor.

16 Note that we have not tried to optimise constants in the above proof. For example, with more work the constant 14 can be reduced to less than 3.

Finally, we address the computational complexity of our main theorem.

Theorem 17. There exists a function f : N → N and an algorithm that takes as input an n-vertex graph and outputs, in O(f(k)n) time, an (f(k), f(k))-good tree decomposition of G and a layered path decomposition of G of layered width at most f(k), where k is largest

integer such that G contains a Qk minor.

Proof Sketch. In the following, for each i ∈ {0,..., 5}, fi : N → N is an unspecified function that is known to exist. Since G has no Qk-minor, |E(G)| 6 f0(k)n (see [32]). An SPQR-tree S of G can be computed in O(f0(k)n) time, and the total size of the graphs {Ha : a ∈ V (S)} is at most O(f0(k)n) (see [25]). A path decomposition of Ha with width at most 2 for each S-node or P-node a is easily computed in time linear in the size of Ha. For each R-node a, the pathwidth of Ha is at most p = p(k). Taking p(k) to be part of f1(k), a minimum-width path decomposition of Ha for an R-node a can be computed in O(f1(k)|V (Ha)|) time [3–5, 27]. These path decompositions are all that is needed to construct the tree T and an

(f2(k), f3(k))-good tree decomposition {Ba : a ∈ V (T )} in O(f1(k)n) time. The proof that (5) implies (4) in Section2 is constructive and immediately gives an

O(f4(k)n) time algorithm to convert the (f2(k), f3(k))-good tree decomposition into a layered path decomposition of layered width at most f5(k). This establishes the theorem for f(k) = max{fi(k): i ∈ {0,..., 5}}.

Acknowledgements

Thanks to the anonymous referees for pointing out some missing details and for several other helpful comments. Some of this research took place at the 2018 Graphs@IMPA workshop in Rio de Janeiro, February 2018.

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20